Bifurcations of Invariant Tori in Predator-Prey Models with Seasonal Prey Harvesting

In this paper we study bifurcations in predator-prey systems with seasonal prey harvesting. First, when the seasonal harvesting reduces to constant yield, it is shown that various kinds of bifurcations, including saddle-node bifurcation, degenerate Hopf bifurcation, and Bogdanov– Takens bifurcation (i.e., cusp bifurcation of codimension 2), occur in the model as parameters vary. The existence of two limit cycles and a homoclinic loop is established. Bifurcation diagrams and phase portraits of the model are also given by numerical simulations, which reveal far richer dynamics compared to the case without harvesting. Second, when harvesting is seasonal (described by a periodic function), sufficient conditions for the existence of an asymptotically stable periodic solution and bifurcation of a stable periodic orbit into a stable invariant torus of the model are given. Numerical simulations, including bifurcation diagrams, phase portraits, and attractors of Poincaré maps, are carried out to demonstrate the existence of bifurcation of a stable periodic orbit into an invariant torus and bifurcation of a stable homoclinic loop into an invariant homoclinic torus, respectively, as the amplitude of seasonal harvesting increases. Our study indicates that to have persistence of the interacting species with seasonal harvesting in the form of asymptotically stable periodic solutions or stable quasi-periodic solutions, initial species densities should be located in the attraction basin of the hyperbolic stable equilibrium, stable limit cycle, or stable homoclinic loop, respectively, for the model with no harvesting or with constant-yield harvesting. Our study also demonstrates that the dynamical behaviors of the model are very sensitive to the constant-yield or seasonal prey harvesting, and careful management of resources and harvesting policies is required in the applied conservation and renewable resource contexts.